We prove the consistency of: if B1 B2 are Boolean algebras satisfying the c.c.c. and the 2אo-c.c. respectively then B1 × B2 satisfies the 2אo-c.c. We start with a universe with a Ramsey cardinal (less suffice).
Shelah, S. (1996). Was Sierpiński right? III can continuum-c.c. Times c.c.c. be continuum-c.c.? Annals of Pure and Applied Logic, 78(1–3), 259–269. https://doi.org/10.1016/0168-0072(95)00036-4